386 



BELL SYSTEM TEC/LMCAL JOIKNAL 



For the example taken in the text, when the pulse train was subdivided 

 into 10 subsiding pulse trains, the period T = 1/v = 10/c = 40L. Thus in 

 this case, the Fourier coefficients of the harmonics of v are 



2E . TTii / AL\ 



(5a) 



The expression for .1,, in equation (5) can be put in simpler form by using 

 the formula for the sin of tlie sum of t wo angles. In this way, we get 



An — 



IE 



irn 



TTll 



sm I — I cos 



/irn AL 



L\ , /7r//\ . /irn AL 



(6) 



Now, for // odd, sia — alternately assumes the value ± 1 and cos — vanishes. 



(?) 



and for ii even, cos ( — - ) alternatelv assumes the value ±1 and sin 



irn 



vanishes. The A o term, being the d-c average of the pulse train, is given by 



E/2{L + AL) ^E (. , AL 

 T 2 V T 



(7) 



If the pulse train is transformed by shifting the zero so that it alternates 

 between db£/2 instead of and E, the first term in equation (7) vanishes 

 and (2) becomes, from (6) & (7), 



e(t) = Ao A- Ai cos 27rf/ 

 + Ai cos 2x 2cl + • 



Where 



etc. 



A, = 



A. = 



m 



2E /t 



1 = — cos ( - 



TT \Z 



¥) 



2L; . ML 

 ^^ = 2. "" " U 



A, = 



2E Stt /AL\ 



3. ^^^ T \-l) 



(8) 



APPENDIX D 



The purjjose of this section is to comi)ule the si)ectrum of the carrier given 

 by e(|ualion (S) in A])pendix C as their amplitudes vary with - = k sin vl. 



